In this paper we present an isogeometric formulation for rotation-free thin shell analysis of structures comprised of multiple patches. The structural patches are
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1- or higher-order continuous in ...the interior, and are joined with
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0-continuity. The Kirchhoff–Love shell theory that relies on higher-order continuity of the basis functions is employed in the patch interior as presented in Kiendl et al. 36. For the treatment of patch boundaries, a method is developed in which strips of fictitious material with unidirectional bending stiffness and zero membrane stiffness are added at patch interfaces. The direction of bending stiffness is chosen to be transverse to the patch interface. This choice leads to an approximate satisfaction of the appropriate kinematic constraints at patch interfaces without introducing additional stiffness to the shell structure. The attractive features of the method include simplicity of implementation and direct applicability to complex, multi-patch shell structures. The good performance of the bending strip method is demonstrated on a set of benchmark examples. Application to a wind turbine rotor subjected to realistic wind loads is also shown. Extension of the bending strip approach to the coupling of solids and shells is proposed and demonstrated numerically.
An isogeometric finite element method is presented for natural frequencies analysis of thin plate problems of various geometries. Non-Uniform Rational B-Splines (NURBS) basis function is applied for ...approximation of the thin plate deflection field, as for description of the geometry. The governing and discretized equation for the free vibration of the Kirchhoff thin plates is obtained using the standard Galerkin method. Several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of proposed method and compared with the theoretical solutions and other numerical methods.
► The NURBS-based isogeometric method is developed for eigen-value analysis of plates. ► Several numerical simulations of thin plates with various shapes are examined. ► For such eigen-value problems considering various aspect ratios. ► Proposed approach can yield highly accurate solutions of the eigenvalue problems. ► The proposed method is potential and applicable to practical problems of engineering
We present isogeometric shape optimization for shell structures applying sensitivity weighting and semi-analytical analysis. We use a rotation-free shell formulation and all involved geometry models, ...i.e., initial design, analysis model, optimization model, and final design use the same geometric basis, in particular NURBS. A sensitivity weighting scheme is presented which eliminates certain effects of the chosen discretization on the design update. A multilevel design approach is applied such that the design space can be chosen independently from the analysis space. The use of semi-analytical sensitivities allows having different polynomial degrees for design and analysis model. Different numerical examples are performed which confirm the applicability of the proposed method. Furthermore, a shape optimization example with an exact solution is presented which can serve as general benchmark for shape optimization methods.
We present numerical formulations of Timoshenko beams with only one unknown, the bending displacement, and it is shown that all variables of the beam problem can be expressed in terms of it and its ...derivatives. We develop strong and weak forms of the problem. The strong form of the problem involves the fourth derivative of the bending displacement, whereas the symmetric weak form involves, somewhat surprisingly, third and second derivatives. Based on these, we develop isogeometric collocation and Galerkin formulations, that are completely locking-free and involve only half the degrees of freedom compared to standard Timoshenko beam formulations. Several numerical tests are presented to demonstrate the performance of the proposed formulations.
•We present Timoshenko beam formulations with only one unknown variable.•We introduce the bending displacement as new variable.•Strong and weak forms of the problem are developed.•The problems are solved by isogeometric Galerkin and collocation formulations.•The presented numerical methods are completely locking-free ab initio.
► An isogeometric shell element based on the thin shell theory is proposed. ► The PHT-spline tremendously facilitates local refinement and possess
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1 continuity. ► The thin shell analysis based on ...Kirchhoff–Love theory avoids the use of rotational degrees of freedom.
This paper presents a novel approach for isogeometric analysis of thin shells using polynomial splines over hierarchical T-meshes (PHT-splines). The method exploits the flexibility of T-meshes for local refinement. The main advantage of the PHT-splines in the context of thin shell theory is that it achieves
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1 continuity, so the Kirchhoff–Love theory can be used in pristine form. No rotational degrees of freedom are needed. Numerical results show the excellent performance of the present method.
Within the general framework of isogeometric methods, collocation schemes have been recently proposed as a viable and promising low-cost alternative to standard isogeometric Galerkin approaches. In ...this paper, isogeometric collocation methods for the numerical approximation of Reissner–Mindlin plate problems are proposed for the first time. Locking-free primal and mixed formulations are herein considered, and the potential of isogeometric collocation as a geometrically flexible and computationally efficient simulation tool for shear deformable plates is shown through the solution of several numerical tests.
•Isogeometric collocation methods for Reissner–Mindlin plate problems are proposed for the first time.•Locking-free primal and mixed formulations are proposed.•Numerical tests show the efficiency of the methods.
•We apply isogeometric collocation techniques to spatial Timoshenko rods.•We solve the strong form equations of the problem in both displacement-based and mixed formulations.•We prove that mixed ...collocation schemes are locking-free independently of the polynomial degrees for the unknown fields.•Numerical experiments confirm the accuracy and efficiency of the considered methods.
In this work we present the application of isogeometric collocation techniques to the solution of spatial Timoshenko rods. The strong form equations of the problem are presented in both displacement-based and mixed formulations and are discretized via NURBS-based isogeometric collocation. Several numerical experiments are reported to test the accuracy and efficiency of the considered methods, as well as their applicability to problems of practical interest. In particular, it is shown that mixed collocation schemes are locking-free independently of the choice of the polynomial degrees for the unknown fields. Such an important property is also analytically proven.
We present a reformulation of the classical Timoshenko beam problem, resulting in a single differential equation with the rotation as the only primal variable. We show that this formulation is ...equivalent to the standard formulation and the same types of boundary conditions apply. Moreover, we develop an isogeometric collocation scheme to solve the problem numerically. The formulation is completely locking-free and involves only half the degrees of freedom compared to a standard formulation. Numerical tests are presented to confirm the performance of the proposed approach.
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